Question: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{-4y^2 - 44y - 72}{-6y^2 + 42y + 108}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {-4(y^2 + 11y + 18)} {-6(y^2 - 7y - 18)} $ $ z = \dfrac{4}{6} \cdot \dfrac{y^2 + 11y + 18}{y^2 - 7y - 18} $ Simplify: $ z = \dfrac{2}{3} \cdot \dfrac{y^2 + 11y + 18}{y^2 - 7y - 18}$ Next factor the numerator and denominator. $ z = \dfrac{2}{3} \cdot \dfrac{(y + 2)(y + 9)}{(y + 2)(y - 9)}$ Assuming $y \neq -2$ , we can cancel the $y + 2$ $ z = \dfrac{2}{3} \cdot \dfrac{y + 9}{y - 9}$ Therefore: $ z = \dfrac{ 2(y + 9)}{ 3(y - 9)}$, $y \neq -2$